Viscous and heavy oil subsurface deposits represent a significant portion of the recoverable hydrocarbon reserve in the world. Heavy hydrocarbons cannot be efficiently recovered by the conventional oil recovery techniques (primary and secondary) because of relatively high viscosity and therefore low mobility of oil. Hot fluid injection is one of the successful techniques that is currently adopted in the industry to reduce oil viscosity and mobilize oil towards the production wells. Numerical methods are widely used in the oil industry as a means to model the mechanisms that dominate fluid flow behavior in the subterranean formation. Computer simulations help to predict reservoir performance with different scenarios that are intended to optimize recovery processes and the corresponding economic forecast.
In reservoir simulation, numerical methods are used to approximate the solution of the mathematical equations that describe the material balance and dynamic behavior of multiphase, multicomponent fluid flow in the subsurface. The simulation model predicts the thermodynamic behavior of several hydrocarbon components under certain temperature and pressure conditions, the interaction between the fluids and the rock formation, and rock mechanics. Reservoir simulation, in a larger sense, coordinates the underground transient flow behavior with the surface processing facilities that manage the injection and production rates in the wells and the surface flowlines constraints.
Two types of simulation models are common in reservoir simulation literature: compositional and black oil. In a compositional model, the number of components and pseudo-components is typically around ten and the thermodynamic phase behavior is usually modeled by an equation of state (EOS). The EOS predicts the phase split of a mixture into gas and oil phases and estimates the compositions of each phase. The black-oil model is a simplification of the compositional model. It incorporates simulation of three components that correspond to gas, oil, and water phases.
In conventional oil recovery models, temporal and spatial variations of the temperature in the reservoir are usually negligible. The system is considered isothermal and therefore solving the energy equation is not needed. In thermal recovery processes, however, the energy equation should be solved in conjunction with the flow equations.
Simulation models require input data that describe reservoir geometry, rock properties such as porosity and permeability, fluid properties such as fluid composition, and pressure-volume-temperatures (PVT) information of the fluid, and well production and injection data.
Finite difference (FD) is one of the numerical methods that is mostly used in commercial reservoir simulators. In this method, the reservoir geometry is subdivided into a grid composed of contiguous and non-overlapping volume entities known as grid-cells or grid-blocks. Two grid-types are commonly used in reservoir simulation literature: regular Cartesian grid and irregular corner-point-geometry grid. Rock properties are assigned to each grid-block and the sought variables such as the pressure, phase saturations and composition are calculated as average values in the grid-blocks. The number of grid-blocks in a simulation model depends on the desired resolution of the solution, the size of the reservoir, and the level of geological complexities, such as number of faults and rock heterogeneities.
In the FD scheme, the Taylor series expansion is used to define the derivative functions in governing flow and energy equations. Most commercial models use the first order form of the approximation of derivatives. As a result, state variables such as saturation, composition and temperature are computed to be constant in a computational grid-block. There are a few inherent advantages of the finite-difference method including: 1) simplicity; 2) ease of extension from 1D to 2D and 3D; and 3) compatibility with certain aspects of physics of two- and three-phase flow. On the other hand, one of the major disadvantage of the FD method is that it provides poor accuracy if the solution has sharp changes in space such as in case of moving heat front in hot fluid injection process. The FD method may introduce significant numerical dispersion that smears sharp fronts in the solution. An assessment of numerical dispersion influence in isothermal compositional modeling is provided by Coats “An Equation of State Compositional Moder’ (October 1980, Society of Petroleum Engineering), pp. 363-376. In hot fluid injection processes in heavy oil reservoirs, temperature has significant influence on oil viscosity and consequently on the ultimate oil recovery prediction. Accurate prediction of the heat front is therefore crucial. The need for fine gridding in thermal recovery models, such as steam-assisted-gravity-drainage (SAGD) is shown by Card et al. “Numerical Modeling of Advanced In-Situ Recovery Processes in Complex Heavy-Oil and Bituman Reservoirs” (November 2005, Society of Petroleum Engineering, SPE97476). The SAGD process is described in the Canadian patent 1,304,287.
The FD method may require an excessive number of grid-blocks to improve the accuracy of the solution, which eventually may add significant computation time. U.S. Pat. No. 7,164,990 B2 uses a streamline method to reduce numerical dispersion in the FD method. Dynamic grid refinement is another technique suggested in the literature to reduce the number of grid-blocks in unwanted regions in the reservoir. One embodiment of Dynamic grid refinement is described by Sammon (Dynamic Grid Refinement and Amalgamation for Compositional simulation” (February 2003, Society of Petroleum Engineering, SPE79683).